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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 795358, 12 pages
Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays
1Department of Science, Bengbu College, Bengbu 233030, China
2Department of Mechanical and Electronic Engineering, Bengbu College, Bengbu 233030, China
3Faculty of Science, Jiangsu University, Zhenjiang 212013, China
Received 5 May 2013; Accepted 13 May 2013
Academic Editor: Luca Guerrini
Copyright © 2013 Juan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.
Multispecies predator-prey models have been studied by many scholars [1–7]. Guo and Jiang  studied the following three-species food-chain system: where , , and are the population densities of the prey, the predator and the top predator at time . The prey grows with intrinsic growth rate and carrying capacity in the absence of predation. The predator captures the prey with capture rate and Holling type II functional response . The top predator captures its prey (the predator) with capture rate and Holling type I functional response . The predator and the top predator contribute to their growth with the conversion rates and , respectively. The parameters and are the death rates of the predator and the top predator, respectively. All the parameters , , , , , , , , and in system (1) are assumed to be positive. The constant represents the time delay due to the gestation of the prey. Guo and Jiang  investigated the bifurcation phenomenon and the properties of periodic solutions of system (1).
Predator-prey systems with single delay as system (1) have been investigated extensively [8–12]. However, there are some papers on the bifurcations of a population dynamics with multiple delays [13–16]. Gakkhar and Singh  studied the effects of two delays on a delayed predator-prey system with modified Leslie-Gower and Holling type II functional response and established the existence of periodic solutions via Hopf bifurcation with respect to both delays. Motivated by the work of Guo and Jiang  and Gakkhar and Singh , we consider the following predator-prey system with two delays: where denotes the time delay due to the gestation of the predator and denotes the time delay due to the gestation of the top predator.
This paper is organized as follows. In the next section, we will consider the stability of the positive equilibrium of system (2) and the existence of local Hopf bifurcation at the positive equilibrium. In Section 3, we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions from the Hopf bifurcation. Some numerical simulations are also given to illustrate the theoretical prediction in Section 4.
2. Local Stability and Hopf Bifurcation
Because we are only interested in the case in which the species can coexist, then we only consider the positive equilibrium of system (2). It is not difficult to know that if conditions and hold, then system (2) has a unique positive equilibrium , where where
Let , , and and still denote , , and by , , and respectively. Then system (2) can be transformed to the following form: where The linearized system of system (5) is Then the associated characteristic equation of system (7) at the origin is of the form where
Case 1. One has .
Equation (8) becomes Obviously, if conditions and hold, then all the roots of (10) must have negative real parts. Then, we can conclude that the positive equilibrium is locally asymptotically stable in the absence of delay.
Case 2. One has .
Equation (8) becomes Letting be a root of (11), then we have It follows that where Letting , then (13) becomes Obviously, . Thus, we assume that (15) has at least one positive solution. Without loss of generality, we assume that it has three positive roots, which are denoted as , , and . Then (13) has three positive roots , .
From (12), we can get Then, we denote Next, we verify the transversality condition. Differentiating the two sides of (11) with respect to and noticing that is a function of , we can get Therefore From (13), we have Therefore with Obviously, if , then . Thus, if condition holds, the transversality condition is satisfied. In conclusion, we have the following results.
Case 3. One has .
Equation (8) becomes Let be a root of (23), then we have which follows that with Let , then (24) becomes Similar as in Case 2, we assume that (27) has at least one positive solution. Without loss of generality, we assume that it has three positive roots, which are denoted by , and . Then (25) has three positive roots , .
From (24), we get Then, we denote Similar as in Case 2, we know that if condition holds, where then, . Namely, if condition holds, the transversality condition is satisfied. Therefore, we have the following results. Therefore, we have the following theorem.
Case 4. One has .
It is considered that with (8), in its stable interval and is considered as a parameter.
Let be the root of (8). Separating real and imaginary parts leads to Eliminating leads to where Suppose that (32) has finite positive roots. If condition holds, we denote the roots of (32) by . For every fixed , there exists a sequence satisfying (32).
To verify the transversality condition of Hopf bifurcation, differentiating (8) with respect to and substituting , we can get where Clearly, if condition holds, then . Namely, if condition holds, the transversality condition is satisfied. Therefore, we have the following results. Thus, we have the following theorem.
Case 5. One has .
We consider (8) with in its stable interval, regarding as a parameter.
Let be a root of (8). Then we get It follows that where Similar as in Case 4, we suppose that (39) has finite positive roots. And we denote the roots of (39) by . The corresponding critical value of is Let when (8) has a pair of purely imaginary roots for .
Similar as in Case 4, we give the following assumption , where
Therefore, if condition holds, then we can get . That is, the transversality condition is satisfied. Hence, we have the following theorem.
3. Direction and Stability of the Hopf Bifurcation
In this section, we will employ the normal form method and center manifold theorem introduced by Hassard et al.  to determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system (2) with respect to for . Without loss of generality, we assume that , where .
Let . Then is the Hopf bifurcation value of system (2). Rescaling the time delay , then system (2) can be rewritten as where With By Riesz representation theorem, there exists a matrix function whose elements are of bounded variation, such that In fact, we can choose For , we define Then system (44) can be transformed into the following operator equation:
The adjoint operator of is defined by associated with a bilinear form where .
From the above discussion, we know that are the eigenvalues of and they are also eigenvalues of . We assume that is the eigenvector belonging to the eigenvalue and is the eigenvector belonging to the eigenvalue . Then, by a simple computation, we can obtain Then we have .
Next, we get the coefficients used to determine the important quantities of the periodic solution by using a computation process similar to that in : with where and can be computed as the following equations, respectively with Thus, we can calculate the following values: Based on the discussion above, we can obtain the following results.
Theorem 5. If , then the Hopf bifurcation is supercritical (subcritical); if , the bifurcating periodic solutions are stable (unstable); if , the period of the bifurcating periodic solutions increases (decreases).
4. Numerical Simulation and Discussion
In this section, we present some numerical simulations to illustrate the analytical results obtained in the previous sections. Let , , , , , , , , and . Then we have the following particular case of system (2): which has a positive equilibrium .
For , we have , . From Theorem 1, we know that the positive equilibrium is asymptotically stable for . As can be seen from Figure 1, if , is asymptotically stable. However, if , then is unstable and system (59) undergoes a Hopf bifurcation at , and a family of periodic solutions bifurcate from the positive equilibrium . This property can be illustrated by Figure 2. For , by a simple computation, we can easily get , . The corresponding waveform and the phase plots are shown in Figures 3 and 4.
For and , we get , . That is, when increases from zero to the critical value , the positive equilibrium is asymptotically stable; then it will lose stability, and a Hopf bifurcation occurs once . This property can be illustrated by Figures 5 and 6. Further, we get , . Then we have , , . Therefore, from Theorem 5, we can know that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable.
Guo and Jiang  have obtained that the three species in system (2) with only one time delay can coexist, however, we get that the species could also coexist with some available time delays of the predator and the top predator. This is valuable from the view of ecology. As the future work, we shall consider the following more general and more complicated system with multiple delays: where is feedback delay of the prey and , are the time delays due to the gestation of the predator and the top predator, respectively.
This work was supported by National Natural Science Foundation of China (11072090), Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2013B137) and Anhui Provincial Natural Science Foundation under Grant no. 1208085QA11.
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